Section 1.5 : Solving Trig Equations with Calculators, Part I
6. Find the solution(s) to \(\displaystyle 3\tan \left( {\frac{w}{4}} \right) - 1 = 11 - 2\tan \left( {\frac{w}{4}} \right)\) that are in \(\left[ { - 50,0} \right]\). Use at least 4 decimal places in your work.
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Isolating the tangent (with a coefficient of one) on one side of the equation gives,
\[\tan \left( {\frac{w}{4}} \right) = \frac{{12}}{5}\]First, using our calculator we can see that,
\[\frac{w}{4} = {\tan ^{ - 1}}\left( {\frac{{12}}{5}} \right) = 1.1760\]As we discussed in Example 5 of this section the second angle for equations involving tangent will always be the \(\pi \) plus the first angle. Therefore, \(\pi + 1.1760 = 4.3176\) will be the second angle.
From the discussion in the notes for this section we know that once we have these two angles we can get all possible angles by simply adding “\( + \,2\pi n\) for \(n = 0, \pm 1, \pm 2, \ldots \)” onto each of these.
This then means that we must have,
\[\frac{w}{4} = 1.1760 + 2\pi n\hspace{0.25in}{\mbox{OR }}\hspace{0.25in}\frac{w}{4} = 4.3176 + 2\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots \]Finally, to get all the solutions to the equation all we need to do is multiply both sides by 4 and we’ll convert everything to decimals to help with the final step.
\[\begin{align*}w & = 4.7040 + 8\pi n & \hspace{0.25in}{\mbox{OR }}\hspace{0.25in} & w = 17.2704 + 8\pi n\hspace{0.25in} & n = 0, \pm 1, \pm 2, \ldots \\ & = 4.7040 + 25.1327n & \hspace{0.25in}{\mbox{OR }}\hspace{0.25in} & \hspace{0.15in} = 17.2704 + 25.1327n\hspace{0.25in} & n = 0, \pm 1, \pm 2, \ldots \end{align*}\]Now let’s find all the solutions. First, notice that if we plug in positive \(n\) or \(n = 0\) we will have positive solutions and these solutions will be out of the interval. Therefore, we’ll start with \(n = - 1\).
\[\begin{array}{lclcl}{n = - 1:\,}&{w = - 20.4287\,}& \hspace{0.25in} {{\mbox{OR}}} \hspace{0.25in} &{w = - 7.8623}\\{n = - 2:}&{w = - 45.5614\,}& \hspace{0.25in} {{\mbox{OR}}} \hspace{0.25in} &{\hspace{0.10in} w = - 32.9950}\end{array}\]Notice that with each increase in \(n\) we were really just subtracting another 25.1327 from the previous results. A quick inspection of the results above will quickly show us that we don’t need to go any farther and we won’t bother with any other values of \(n\).
So, it looks like we have the four solutions to this equation in the given interval.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{w = - 45.5614,\,\, - 32.9950,\,\, - 20.4287,\,\, - 7.8623}}\]Note that depending upon the amount of decimals you used here your answers may vary slightly from these due to round off error. Any differences should be slight and only appear around the 4th decimal place or so however.